Finite Group Databasebeta

← #145397

Subgroups — #145397

Order 512.

318 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1670C211yesmaximal, characteristic
2#13263C211yesmaximal, characteristic
3#13203C211yesmaximal, characteristic
4C2×#1372C211yesmaximal, characteristic
5C2×#1362C211yesmaximal, characteristic
6#8645C211yesmaximal, characteristic
7#7274C211yesmaximal, characteristic
8C2×#494C2211yescharacteristic
9C2×#475—21no—
10C2×C8×Q8C2211yescharacteristic
11C2×C4⋊C16C2211yescharacteristic
12C2×#358—21no—
13C2×#358—21no—
14C2×C4×C16C2211yescharacteristic
15#1845C2211yescharacteristic
16#1803—21no—
17#1670—22no—
18#1670—22no—
19C2×#335C2211yescharacteristic
20C2×#335C2211yescharacteristic
21C2×#332—21no—
22#1374—21no—
23#1374—21no—
24#1372—22no—
25#1362—22no—
26#1296—21no—
27C22×Q16—21no—
28C22×Q16—21no—
29C2×C4⋊Q8—21no—
30C2×C4×Q8D411yescharacteristic
31C22×C16—21no—
32C22×C16D411yescharacteristic
33C22×C16D411yescharacteristic
34#494—21no—
35#494—21no—
36#494—21no—
37#494D412yes—
38#475—24no—
39#475—42no—
40C8×Q8—21no—
41C8×Q8—21no—
42C8×Q8—21no—
43C8×Q8—21no—
44C2×#107—21no—
45C2×#107—21no—
46C2×#106—41no—
47C2×C4⋊C8—21no—
48C2×C4⋊C8—21no—
49C2×C4⋊C8—21no—
50C2×C4⋊C8D411yescharacteristic
51C2×#103—41no—
52C2×#103—41no—
53C2×C4×C8—21no—
54C2×C4×C8C2311yescharacteristic
55#384—21no—
56C4⋊C16—21no—
57C4⋊C16—21no—
58#358—42no—
59#358—42no—
60C4×C16—21no—
61C4×C16—21no—
62#340—41no—
63#335—21no—
64#335—21no—
65#335—21no—
66#335D412yes—
67#332—24no—
68C22×Q8—41no—
69C22×Q8D811yescharacteristic
70C2×Q16—22no—
71C2×Q16—22no—
72C2×Q16—41no—
73C2×Q16—41no—
74C2×Q16—41no—
75C2×Q16—41no—
76C2×Q16—41no—
77C2×Q16—41no—
78C2×Q16—81no—
79C2×Q16—81no—
80C22×C8—21no—
81C22×C8—41no—
82C22×C8—41no—
83C22×C8C2×D411yescharacteristic
84C22×C8#4111yescharacteristic
85C4⋊Q8—22no—
86C4⋊Q8—41no—
87C4⋊Q8—41no—
88C4⋊Q8—41no—
89C4×Q8—21no—
90C4×Q8—21no—
91C4×Q8—21no—
92C4×Q8—21no—
93C2×C4⋊C4—21no—
94C2×C4⋊C4—41no—
95C2×C4⋊C4—41no—
96C2×C4⋊C4—41no—
97C2×C4⋊C4—41no—
98C2×C4⋊C4—41no—
99C2×C4⋊C4D811yescharacteristic
100C2×C42—21no—
101C2×C42C2×D411yescharacteristic
102C2×C16—21no—
103C2×C16—21no—
104C2×C16—21no—
105C2×C16—21no—
106C2×C16—21no—
107C2×C16—41no—
108C2×C16—41no—
109C2×C16—41no—
110C2×C16SD1612yes—
111#107—41no—
112#107—41no—
113#107—81no—
114#106—44no—
115C4⋊C8—21no—
116C4⋊C8—22no—
117C4⋊C8—21no—
118C4⋊C8—41no—
119C4⋊C8—41no—
120C4⋊C8—41no—
121C4⋊C8—41no—
122C4⋊C8—41no—
123#103—44no—
124#103—82no—
125C4×C8—41no—
126C4×C8—41no—
127C4×C8C2×D411yescharacteristic
128C4×C8C2×D411yescharacteristic
129C4×C8#4111yescharacteristic
130C4×C8#4111yescharacteristic
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—42no—
137C2×Q8—81no—
138C2×Q8—81no—
139C2×Q8—81no—
140C2×Q8—81no—
141C2×Q8—81no—
142C2×Q8SD3212yes—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—81no—
146C22×C4—81no—
147C22×C4#12111yescharacteristic
148C22×C4C2×D811yescharacteristic
149C22×C4#13511yescharacteristic
150Q16—42no—
151Q16—42no—
152Q16—82no—
153Q16—82no—
154Q16—81no—
155Q16—81no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8—41no—
159C2×C8—42no—
160C2×C8—41no—
161C2×C8—41no—
162C2×C8—81no—
163C2×C8—81no—
164C2×C8—81no—
165C2×C8—81no—
166C2×C8—81no—
167C2×C8#12111yescharacteristic
168C2×C8#12111yescharacteristic
169C2×C8#12311yescharacteristic
170C2×C8#12411yescharacteristic
171C2×C8C2×SD1611yescharacteristic
172C2×C8#13511yescharacteristic
173C2×C8AGL(1,ℤ/8)11yescharacteristic
174C2×C8#13711yescharacteristic
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4—81no—
182C4⋊C4—81no—
183C4⋊C4—81no—
184C4⋊C4—81no—
185C4⋊C4—81no—
186C4⋊C4—81no—
187C4⋊C4—161no—
188C42—41no—
189C42—41no—
190C42#12011yescharacteristic
191C42#12311yescharacteristic
192C42AGL(1,ℤ/8)11yescharacteristic
193C42#13711yescharacteristic
194C16—22no—
195C16—41no—
196C16—81no—
197C23#46611yescharacteristic
198Q8—22no—
199Q8—22no—
200Q8—41no—
201Q8—41no—
202Q8—82no—
203Q8—82no—
204Q8—161no—
205Q8—161no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—82no—
215C2×C4—161no—
216C2×C4—161no—
217C2×C4—161no—
218C2×C4—161no—
219C2×C4—161no—
220C2×C4#39411yescharacteristic
221C2×C4#44711yescharacteristic
222C2×C4#45211yescharacteristic
223C2×C4#46011yescharacteristic
224C2×C4#46411yescharacteristic
225C2×C4#46811yescharacteristic
226C2×C4#47111yescharacteristic
227C2×C4#47211yescharacteristic
228C2×C4#48011yescharacteristic
229C2×C4#48211yescharacteristic
230C2×C4C2×SD3211yescharacteristic
231C2×C4#50811yescharacteristic
232C2×C4#51011yescharacteristic
233C2×C4#51011yescharacteristic
234C8—21no—
235C8—21no—
236C8—81no—
237C8—82no—
238C8—161no—
239C8#46111yescharacteristic
240C8#46311yescharacteristic
241C8#46511yescharacteristic
242C8#46811yescharacteristic
243C22#166811yescharacteristic
244C22#168211yescharacteristic
245C22#203311yescharacteristic
246C22#221711yescharacteristic
247C22#221911yescharacteristic
248C22#222211yescharacteristic
249C22#222311yescharacteristic
250C4—21no—
251C4—21no—
252C4—42no—
253C4—81no—
254C4—162no—
255C4—321no—
256C4#168611yescharacteristic
257C4#169811yescharacteristic
258C4#205111yescharacteristic
259C4#205511yescharacteristic
260C4#218511yescharacteristic
261C4#218911yescharacteristic
262C4#223511yescharacteristic
263C4#223711yescharacteristic
264C2#1047411yescharacteristic
265C2#1244111yescharacteristic
266C2#1244311yescharacteristic
267C2#1247811yescharacteristic
268C2#1248111yescharacteristic
269C2#1338211yescharacteristic
270C2#1339411yescharacteristic
subgroupG/H
center, socle, prime radicalC23#466
commutator, Frattini, cosocleC2×C4×C8C23
LCSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#16702
#132632
#132032
C2×#13722
C2×#13622
#86452
#72742